integral of 6x^2sin(3x)

解答

\int 6 x^{2} sin (3 x) d x

6 (- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x))) + C

求解步骤

\int 6 x^{2} sin (3 x) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int x^{2} sin (3 x) d x

使用分布积分法

= 6 (- \frac{1}{3} x^{2} cos (3 x) - \int - \frac{2}{3} x cos (3 x) d x)

\int - \frac{2}{3} x cos (3 x) d x = - \frac{2}{27} (3 x sin (3 x) + cos (3 x))

= 6 (- \frac{1}{3} x^{2} cos (3 x) - (- \frac{2}{27} (3 x sin (3 x) + cos (3 x))))

化简

= 6 (- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x)))

解答补常数

= 6 (- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x))) + C

n = 1 \sum \infty 2 - \frac{7}{8 ^{n}}

\int 3^{x + 1} d x

h \to 0 lim (\frac{1}{h})

\int \frac{1}{1 + e ^{4 x}} d x

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